Tangible Experience - Regression Line (TLS and OLS)

The GeoGebra applet is for the concepts of Total Least Squares (TLS) or Orthogonal Regression; however, the formula is based on the OLS method

Minimizing the Shortest Distance If we want to minimize the square of the shortest (perpendicular) distances from each data point to the regression line, this approach is known as: Total Least Squares (TLS) or Orthogonal Regression

In TLS, errors are assumed in both the x- and y-values.
The line is adjusted to minimize the perpendicular distance between the observed data and the regression line.
This approach is common in:

Physics (e.g., fitting data with measurement errors in both variables).
Signal processing.
Geometric modeling.

The video lesson below captures the concepts of Ordinary Least Squares (OLS) method. OLS Minimizes Vertical Distances

Ordinary Least Squares (OLS) minimizes the sum of the squared vertical distances (yi−y^i)
This makes sense when the x-values are considered error-free, and we're modeling y as a function of x.

The formula in the GeoGebra applet is based on the OLS method.

Optimizing the Regression Line Using OLS: Minimizing Vertical Distances

How to Use This GeoGebra Build for the OLS Method?

1.) Considering the set of data points in blue and the red line, which serves as an estimator of the best fit or regression line, try to guess the optimized slope and optimized y-intercept using the sliders. (Hint: The regression line is the magenta line.) 2.) As you manually derive the best fit or regression line, observe and compare the non-optimized sum of squares (component a) with the least sum of squares (component o). Reflect on how the formula for the regression line is derived. 3.) Study the Algebra pane section and try to find the command for finding or graphing the regression line given a set of data points. 4.) Study the spreadsheet and try to replicate this build in order to experience the concepts of the regression line firsthand. 5.) Finally, for the beautiful symphony of algebra and calculus, watch the video below for the formal derivation of the regression line equation. Simplify the optimized formula for the slope and y-intercept (b) in order to derive the formula for the optimized slope and y-intercept (b) found in the GeoGebra applet above.

Derive the Formulae for Simple Linear Regression

Red line

1. What does the red line in the GeoGebra build represent?

Select all that apply

A
A) The data points
B
B) The manually estimated regression line
C
C) The regression line or best fit line
D
D) The algebraic equation of the line

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Estimates vs Optimized Values of Slope and Y-intercept

2. When deriving the regression line manually, what should be compared?

Select all that apply

A
A) The slope with the y-intercept
B
B) The non-optimized sum of squares and the least sum of squares
C
C) The x-values with the y-values
D
D) The data points with the algebraic equation

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Derivation of the Formulas of the Regression Line

In the context of the regression line equation, what is expected after simplifying the optimized formula of the slope and y-intercept (b)?

Select all that apply

A
A) To graph the line
B
B) To find the non-optimized sum of squares
C
C) To derive the formula for the optimized slope and y-intercept
D
D) To calculate the sum of the data points

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Regression line and the data points

How does the regression line relate to the data points in the scatter plot?

Select all that apply

A
A) It is a straight line that connects all the data points.
B
B) It represents the best linear approximation of the relationship between the variables.
C
C) It passes through the mean of the data points but does not represent the relationship.
D
D) It only passes through the first and last data points.

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Non-optimized sum of squares and the Least sum of squares

What is the significance of comparing the non-optimized sum of squares with the least sum of squares in the context of regression analysis?

Select all that apply

A
A) To determine the most accurate data points
B
B) To minimize the difference between the data points and the regression line
C
C) To estimate the slope of the regression line
D
D) To check the strength of the correlation between the variables

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y-intercept (c) in the regression line equation

What is the role of the y-intercept (c) in the regression line equation?

Select all that apply

A
A) It represents the slope of the regression line.
B
B) It indicates the point where the regression line crosses the x-axis.
C
C) It represents the predicted value of y when x=0.
D
D) It shows the strength of the correlation between x and y.

Check my answer (3)

Tangible Experience - Regression Line (TLS and OLS)

The GeoGebra applet is for the concepts of Total Least Squares (TLS) or Orthogonal Regression; however, the formula is based on the OLS method

Optimizing the Regression Line Using OLS: Minimizing Vertical Distances

How to Use This GeoGebra Build for the OLS Method?

Derive the Formulae for Simple Linear Regression

Red line

Estimates vs Optimized Values of Slope and Y-intercept

Derivation of the Formulas of the Regression Line

Regression line and the data points

Non-optimized sum of squares and the Least sum of squares

y-intercept (c) in the regression line equation

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